Automorphisms of generic gradient vector fields with prescribed finite symmetries

Let $M$ be a compact and connected smooth manifold endowed with a smooth action of a finite group $\mathrm{\Gamma }$, and let $f$ be a $\mathrm{\Gamma }$-invariant Morse function on $M$. We prove that the space of $\mathrm{\Gamma }$-invariant Riemannian metrics on $M$ contains a residual subset ${\mathcal{M}et}_f$ with the following property. Let $g\in {\mathcal{M}et}_f$ and let ${\mathrm{\nabla }}^{g}f$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $\varphi$ of $M$ preserving ${\mathrm{\nabla }}^{g}f$ there exists some real number $t$ and some $\gamma \in \mathrm{\Gamma }$ such that for every $x\in M$ we have $\varphi (x)=\gamma {\mathrm{\Phi }}_t^g(x)$, where ${\mathrm{\Phi }}_t^g$ is the time-$t$ flow of the vector field ${\mathrm{\nabla }}^{g}f$.